Accelerator driven energy production: a comment

Accelerator driven energy production: a comment

Annals of Nuclear Energy 28 (2001) 1477±1483 Technical note Accelerator driven energy production: a comment S.B. Deg...

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Annals of Nuclear Energy 28 (2001) 1477±1483

Technical note

Accelerator driven energy production: a comment S.B. Degweker * Theoretical Physics Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India Received 28 November 2000; accepted 30 November 2000

Abstract An analytical solution of the burn up equations with a time dependent ¯ux in a molten salt accelerator driven subcritical system (ADSS) is obtained. A simple formula for the time required to reach a given concentration of U233 starting with only Thorium (Th) is deduced. It is shown that the time required for such a reactor to reach full power is much larger than stated in a recent report from Los Alamos. Addition of another naturally occurring fuel, viz. natural U could, however, bring down this time to any desired value. # 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Accelerator driven sub-critical systems (ADSS) have attracted worldwide attention in recent years due to their superior safety characteristics and their potential for burning Pu and nuclear waste. A number of concepts, such as molten salt thermal reactor (Bowman, 1997), Pb cooled fast reactors (Rubbia, et al., 1995) and Na cooled fast reactors (Takizuka, 1997) are being examined. While most of these systems assume a large availability of ®ssile material to start these systems, some attention has also been paid to the possibility of starting them without any ®ssile inventory at all. The molten salt concept needs the least amount of ®ssile inventory because of the highest speci®c power among thermal reactors and a low ®ssile fraction in the fuel. A recent report by Bowman (1997) (henceforth referred to as [1]) makes the interesting suggestion that it might be possible to start an accelerator driven sub-critical * Corresponding author. Fax: +91-22-5505150. E-mail address: [email protected] 0306-4549/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(00)00133-X


S.B. Degweker / Annals of Nuclear Energy 28 (2001) 1477±1483

system for energy production, based on a molten salt reactor concept and the 233UTh cycle, without an initial loading of any ®ssile material. It is stated that such a system is expected to produce the ®ssile concentration necessary to attain full power in a matter of six months to one year. In this paper we present a simple analysis which shows that, for the design parameters of the reactor quoted in [1], it would take much longer. While such calculations could be very easily carried out using standard burn up codes, the interesting part of our derivation is that for the pure Th start up situation, we obtain a simple formula to predict the time required for achieving full power in such a reactor, which is also suciently realistic for the situation at hand. We also show that the method can be extended to the case of several actinide species under fairly realistic assumptions. Thus we show that mixing natural uranium with the Th could accelerate the process to make it a more attractive option. 2. Start up with thorium The system considered in [1] requires 10 tons of Th and has a maximum total power of 500 MW(t). It is operated at a Ke€ of 0.95 and is driven by a proton accelerator of 800 MeV energy and a current of 12 mA. The target used is Pb, which, for the stated energy, has a neutron yield of 22 per proton. The ®nal ®ssile (U233) fraction is expected to be 1.1%. A simple calculation shows that the neutron source at the above Ke€ yields a power of 400 MW(t). If one were to take credit for the source importance, a power of 500 MW(t) is possible. How long would it take to reach an enrichment of 1.1%? One could take a simplistic view and assume that all neutrons from the spallation source go into producing ®ssile U233 from the Th, that U233 neither burns nor does it multiply the neutrons. This gives us a time frame of 5.5 years for getting 110 Kg of ®ssile material. However, it may be argued [1] that as the ®ssile material builds up, there would be an increase in the ¯ux and hence in the rate of breeding. To understand this process, we set up the following simple model. We club all losses to the moderator, structures, and leakage into one macroscopic cross-section 1. Fission products are assumed to be continuously removed and hence do not contribute to neutron capture. The number densities of U and Th are denoted by N3 and N2 respectively, and we assume that N2 is practically constant as a function of time. We further assume that the ¯ux is so low that all the Pa233 decays to U233 and that this decay is instantaneous compared to the time scale of interest. The ratio of N3/N2 is denoted by x. We can then easily write down the following equation for N3 dN3 ˆ c2 N2  dt

…c ‡ f †3 N3 '

The ¯ux can be written using the following equation:


S.B. Degweker / Annals of Nuclear Energy 28 (2001) 1477±1483

'ˆ ˆ


S a …1 K†

S f…c ‡ f †3 N3 ‡ c2 N2 ‡ l g…1


where S denotes the external source density and K stands for the e€ective multiplication factor. Writing Kˆ

f3 N3 f…c ‡ f †3 N3 ‡ C2 N2 ‡ l g


in Eq. (2) and substituting the resulting expression for ' in Eq. (1) we obtain (on dividing both sides by N2) dx sfc2 …c ‡ f †3 xg ˆ dt …c ‡ f f †3 x ‡ c2 ‡ l


In this equation we have written s for S/N2, 1 for 1 =N2 and  x for N3/N2. Eq.  (4) can be rewritten in terms of the familiar parameters  ˆ  = …  ‡  † and f3 f c 3    ˆ c2 =…f ‡ c †3 as follows: dx s… x† ˆ dt …1 †x ‡  ‡ l


where we have written l for 1 =…c ‡ f †3 . ;  and l are spectrum dependent and could change with burn up. Assuming that they do not change appreciably, the above equation is readily integrated to give the general solution st ‡ c ˆ …

1†x ‡ ……




where c is a constant of integration. For the system under consideration we determine c using the initial condition that x is zero at t=0. Finally the time t required to reach a ®ssile to fertile ratio x is given by    1 x t ˆ … 1†x ‡ …… 2† l†ln 1 …7† s  This equation gives the time (t) required for reaching the ®ssile to fertile ratio (x) taking into account multiplication in the system and burn up of the ®ssile species. We can also write the expression for the multiplication factor [Eq. (3)] in terms of ;  and l as follows: Kˆ

x x‡‡l



S.B. Degweker / Annals of Nuclear Energy 28 (2001) 1477±1483

According to [1], x=0.011 for attaining full power, and s works out to be 0.0020 per year. For , we use the thermal value, viz. 2.29. The sum  ‡ l is determined from the requirement that K=0.95 and amounts to 0.0155. For obtaining the shortest possible time we set l ˆ 0 which gives  ˆ 0:0155 which is somewhat higher than the thermal value. This could be understood as being due to resonance absorption in Th232. We thus obtain a breeding time of 4.3 years to reach full power. In actual practice, there will be some leakage and losses to the moderator, so l 6ˆ 0 and  will lie between its thermal value of 0.0135 and 0.0155. If we choose the other extreme of using  ˆ 0:0135 and l ˆ 0:0020 we get a breeding time of 5.5 years. Hence we conclude that the time required to reach full power is in the range of 4.3 to 5.5 years, which is much longer than 6 to 12 months as stated in [1]. 3. Mixture of several actinide species Generalisation to several species of fertile and ®ssile nuclides is possible provided we can assume as before that radioactive decay of intermediate species such as Pa233 is instantaneous. As before we assume that each of the fertile species concentration does not change with time and ®ssion products are removed and therefore do not contribute to absorption. To be speci®c, we assume that we have a mixture of Th and natural U salt. In that case there are three ®ssile species, viz. U235, U233 and Pu239 and two fertile, viz. Th and U238. We can add a ®ctitious fertile nuclide corresponding to U235 and set its initial concentration to zero. As before we assume that each of the fertile material concentrations remain constant with time. Denoting the ®ssile and fertile concentrations by Yi and Xi, respectively, we can write down the following equations to describe the process: dYi ˆ …xi Xi dt

yi Yi †'


By changing the independent variable from t to u where 'ˆ

du dt


Eq. (9) becomes dYi ˆ xi Xi du

yi Yi


These are linear di€erential equations for which the following solutions are written down rather easily: Yi ˆ i Xi ‡ …Yi0

i Xi †e

yi u


S.B. Degweker / Annals of Nuclear Energy 28 (2001) 1477±1483


where Yi0 stands for the initial concentration of the ith ®ssile species. To obtain a relation between u and t we use the relation between the ¯ux and the multiplication factor [®rst part of Eq. (2)]. Using the following expression for K, P fyi Yi i P …13† KˆP xi Xi ‡ yi Yi ‡ l i


we ®nally get 'ˆP i

xi Xi ‡

P i

S …yi


fyi †Yi ‡ l

Eq. (10) together with Eqs. (12) and (14) results in the following di€erential equation for u: du P ˆP dt xi Xi ‡ …yi i


S  fyi † i Xi ‡ …Yi0

i Xi †e

yi u

‡ l

Integrating the above equation, we get

Fig. 1. Power and K-e€ variation with time for pure Th.



S.B. Degweker / Annals of Nuclear Energy 28 (2001) 1477±1483

( St ˆ Sl ‡

X i

xi ‡ …1


i †i yi Xi u ‡

X …1

i †…Yi0

i Xi †…1


yi u



…16† The set of Eqs. (12±16) represents a parametric solution for the concentration of various species, the ¯ux, and the multiplication factor as functions of time. In Figs. 1±3 we show the variation of the power and K with time for three di€erent initial ratios of natural U and Th. The ®gures clearly show that the process of ®ssile breeding in a Th fuelled ADSS could be accelerated using another naturally available fuel. Once the reactor reaches full power, the U could be gradually replaced by Th. With a suciently large fraction of natural U to begin with, one can even have full power from day one. We may mention that Bowman has also discussed the possibility of obtaining full power from day one using 20% enriched U instead of natural U. Inclusion of higher actinides in the model is clearly possible since, these equations are linear in the independent variable u. There is, however, a diculty with regard to long lived intermediate products such as Pa233 at high ¯ux. A similar problem is associated with ®ssion products such as Xe135 if these are not removed continuously as is the case in solid fuelled reactors. For both these nuclides, the ¯ux is not eliminated by transforming the independent variable from t to u. For calculating the absorption by these nuclides we can use the equilibrium expressions for the

Fig. 2. Power and K-e€ variation with time for a 25-75 U-Th mixture.

S.B. Degweker / Annals of Nuclear Energy 28 (2001) 1477±1483


Fig. 3. Power and K-e€ variation with time for 50±50 U-Th mixture.

concentration of these nuclides together with the equation for the ¯ux to get a perturbative estimate. However, as regards U233, even the perturbative solution leads to an integral that is not elementary. Another problem is that the perturbative solution is least accurate when it is most signi®cant, viz. when the system is close to critical. 4. Conclusion We have obtained analytical formulae for the time required to reach full power in Th fuelled molten salt ADSS. Our formulae show that starting with Th alone would take much longer than indicated in [1]. However, adding natural uranium in the initial stage could substantially reduce this time. References Bowman, C.D., 1997. Los Alamos National Laboratory ADS Projects in Status Report on ADS, IAEATECDOC-985. Rubbia, C. et al., 1995. Conceptual Design of a Fast Neutron Operated High Power Energy Ampli®er, Report CERN/AT/95-44. Takizuka, T., 1997. JAERI and PNC Ð OMEGA Project in Status Report on ADS, IAEA-TECDOC-985.